By ditching waders and climbing onto special stand-up paddle boards, anglers are accessing new waterways and fresh fish. So what’s the catch?

ALL SUMMER, I’ve been fly fishing for stripers along grassy banks and tidal flats of coastal New Hampshire. I’ve had some luck, but there are times when the fish are maddeningly just out of casting range—too deep to wade, too shallow for a boat, leaving me reaching for beer while the sun sets over marshlands and eager fish chomp everything but my fly.

“That’s exactly why you need a stand-up paddleboard,” said Stephen Ferrell, a Florida fishing guide and an early adopter of a new angling trend that’s migrated from the cool...